Table of Contents Class 1 Review... 1 Fractions... Exponential Expression... 4 Order of Operation ( P-E-M-D-A-S )... 5 Radicals:... 6 Polynomials... 8 Special Products... 9 Factoring... 10 Factoring Binomials... 1 Rationals... 13 Class : Linear Models... 16 Solving Equations... 19 Word Problems... 1 Formulas... 3 Class 3: Quadratics... 6 Quadratic Equation... 8 Completing the Square... 9 Quadratic Formula... 30 Other Equations... 31 Equations with Rational Exponents:... 3 Equations Involving Absolute Values... 33 Inequalities... 33 Absolute Value Inequalities... 34 Class 4: Function and Their Graphs... 35 Graph of a Function... 38 Piece-Wise Functions... 40 Linear Functions and Slope... 41 Class 5: Average Rate of Change and Transformations... 43 Difference Quotient... 43 Basic Functions... 44 Transformations of Functions... 45
Class 6: Composition and Inverse Functions... 47 Algebraic Operations with Functions... 48 Composition of Functions... 49 Inverses... 50 Class 7: Distance and Midpoint Formulas; Circles... 53 Circles... 54 Class 8 & 9 Review and Test 1... 55 Class 10: Angles and Their Measurements... 56 Relationship between Degrees & Radians... 58 Trigonometric Functions... 60 Special Identities... 6 Co-terminal Angles... 63 Class 11: Trigonometric Functions of any Angle... 64 The Signs of the Trigonometric Functions... 65 Reference Angles... 67 Class 1: Trig Functions of Real Numbers & their Graphs... 69 The Graph of Sine... 70 The Graph of Cosine... 7 The Graph of Tangent... 74 Class 13: Inverse Trigonometric Functions & Applications... 75 Angle of Elevation and Angle of Depression... 79 Class 14 Trigonometric Identities... 80 Class 15 Trigonometric Equations... 81 Class 16: The Law of Sines & The Law of Cosines... 83 Class 17 & 18 Review and Test... 86 Class 19: Quadratic Functions... 87 Class 0: Polynomial Functions & Division of Polynomials... 89 Synthetic Division... 91 Zeros of Polynomial Functions... 9 Class 1: Rational Functions... 93 Asymptotes... 94 Characteristics and Graphs of Rational Functions... 95 Variation... 96
Class : Exponential and Logarithmic Functions... 97 Exponential Functions... 98 Compounding... 99 Logarithmic Functions... 100 Natural Logarithm... 101 Logarithmic Functions... 10 Class 3: Exponential and Logarithmic Equations and Logistic Growth... 103 Logistic Growth... 105 Properties of Logarithms... 106 Class 4 & 5 Review and Test 3... 107
Grading Rubric All work in this workbook needs to be in pencil. Completeness 0-1 points -3 points 4-5 points No parts have been Some information is All definitions and completed. Majority written down. Some steps examples are of definitions and or parts are missing. completed, no steps example work is Ordered pairs or scale from missing missing. the graphs are missing. Neatness Organization Correctness The information is not clear or comprehendible. Information is not presented in the right or designated place. There are many mistakes in the work or the definitions. There are some parts that are not clear. Some steps are missing or not clear. Some information is not presented in the designated place or is presented inappropriately. Three or fewer steps don t follow logically. Some equal signs are missing. Three or fewer mistakes in the definitions or worked examples. All work is neatly written and clear. A final answer is circled. All information is in the right and/or designated place. There is a logic flow to all work. No equal or mathematical signs are missing. No mistakes are made and all work is correct. TOTAL 0 Points
Class 1 Review Set: The following are examples of sets Natural numbers: Whole numbers: Integers: Rational numbers Irrational numbers Real numbers Prime number Practice: Give at least three examples of prime numbers Absolute Value of a number a, denoted by a, is Example: 1. 4. 3 1
Fractions A Fraction is Give a few examples Simplifying: To simplify a fraction we Simplify: 3 4 Reciprocal of a fraction is Find the reciprocal of 3 7 Multiplying: To multiply two fractions we Multiply: 11 4 3 Dividing: To divide two fractions we Divide: 11 5 3 6 Adding/Subtracting: To add/subtract fractions we LCD is
Add: 11 5 3 Practice on the following problems: 5 7 3. 11 11 3 5 4. 4 7 5 3 5. 1 0 1 5 6. 8 4 3
Exponential Expression An exponential expression is an expression of the form Give at least two examples: Simplify 3 1 7. 7. 3 9.. 5 4 Exponential Rules: Write the exponential rules for each of the following: x y x 1 a a ab a x a 0 a x a y a b y x a a x a x y Practice: Simplify the following: 6 6 3 r 5x 9x 11. 3 s 9y y 5 1 14. 6 x y (3 x y) 6 4 4 4 7 x y x y 4
Order of Operation ( P-E-M-D-A-S ) Example: Evaluate the following. 16. 3 8 3 7 5(3 4) 5 7 Scientific Notation: A number is written in scientific notation if Give an Example: 17. Practice: Write the number in Scientific Notation 9,060,000,000 = 0.00017 = 18. Write the number in Decimal Notation 3.067 10-4 = 6.00 10 6 = 19. Perform each operation and write in standard form and Scientific Notation 4 810 (9 10 7 )(4 10-9 ) 10 3 5
Radicals: In General: Notation Examples: 5 100 36 1 81 Product Rule: Quotient Rule: Simplify: Simplify: 6 16y 4x 81 4 Add/Subtract: Add the following: 8 63 3 3 7x 54 x 50 3 Multiply Multiply the following: 4 4 3 7 5 8 6
More Definitions: Variables Give at least three examples: Algebraic Expression Give a few examples: We can Evaluate algebraic expressions if we know the value of the variable(s). Example: Evaluate x y if x 3and y Equation: Give two Examples: Solution/Root Check to see if x is asolution to theequation 5x 3 4x 1 7
Polynomials A Polynomial in x is Give at least three examples Fun things we do with polynomials Evaluating: Find the value of the polynomial 6x 11x 0 when x = -1. Simplifying, Adding and Subtracting: Add/Subtract the following: 14y 310 y (5x x 1) ( 6x x 1) Multiplying: Multiply : (5y 6y 7)(4y 3) 8
Special Products FOIL = Foil the following: ( x 7)( x5) Squaring a Binomial: a b ora b Square the following: (x 5) Difference of Squares: a ba b Multiply the following: 1 1 x x 3 3 (x 6 x)(x 6 x) Dividing Polynomials o Dividing by a Monomial:. Divide: 5x 5x 5x 3 o Dividing by a polynomial other than a monomial: Divide: x 7x1 x 3 9
Factoring Factoring is the process of GCF of a list of Integers To find the GCF Findthe GCF(45, 75) Find the GCF of the following numbers: 3and 33 4, and GCF of a list of Common Variables 5 3 Example: Findthe GCF ( x, x ) GCF of a list of a list of TERMS Find the 4 GCF ( 9,15, 6) 3 4 GCF( 9 x,15 x,6 x) SotheGCF ( 9 x,15 x,6 x) 3x 4 GCF ( x, x, x) x 10
The fist step to factoring a polynomial Prime Polynomial Factoring by Grouping is used for To Factor by Grouping Factor the polynomial ab 4a 7b 8 Practice: Factor the following polynomials 1. 5a ab 5a b. 15 xz 15yz 5xy 5y Factoring Trinomials of the form x bx c Example x x 9 0 Practice: Factor the following 3. x 13x x 5x 36 11
Factoring Trinomials of the form ax bx c Example x x 5 1 Factoring Binomials a b a b a b ( )( ) Difference of two squares: Factor a 16 Practice on factoring the following: 4 5x 1 p 81 4 9 48x 3 c 5 a b is Sum/Difference of two Cubes: 3 3 3 3 x y x y Factor the following: 3 3 3 15x 116 p 50 y 1
Rationals Rational Expression Domain To find the domain x 5 Ex : isdefined for x 3, orthe Domain is(, 3) ( 3, ) x 3 Example: Find the domain of the following expressions: 4 x x,, x 1 x 3 6x 5x1 Operations o Simplifying:. Simplify x 6x5 x 5 13
o Multiplying: Multiply x7 x 1 x1 3x1 o Dividing: x x 8 x 4 Divide: x 9 x3 o Adding & Subtracting: Add: x 3 x x x 1 14
Complex Fractions To simplify complex fractions Simplify: 1 3 x 1 3 x 4 Simplify: d d d r r 1 15
Class : Linear Models Objectives: Graph Equations on the Rectangular coordinate system. Solve Linear Equations in One Variable Solve Rational Equations with Variables on the denominator Use Linear Equations to Solve Problems. Solve a Formula for a Variable o The Rectangular Coordinate System comprises of Draw and label a rectangular coordinate system below: The way we plot a point (a,b) in the coordinate system is Practice 1: Plot the following points in the coordinate system below. (4, ), (, -), (-1, -3), (-5, 1), (0, ), (3, 0), (0, -4), (-4, 0) 16
o An equation in two variables such as xy 1 or y= 4x +3 has a solution Graph of an Equation Practice : Graph the equation yx 3 by using the point-plotting method. Practice 3: Graph the equation y x 3 by using the point-plotting method. 17
Practice 4: Graph the equation y x 3 by using the point-plotting method. Intercepts o X-Intercept is o To find the x-intercept o Y -Intercept is o To find the x-intercept Practice 5: Find the x and y-intercepts of x y = 1 18
Solving Equations General Strategy of Solving Linear Equations 1.. 3. 4. 5. 6. Practice: Solve each of the equations 6. x x x 7. 1 3 5 x 4 19
8. 5 x x 7 3 Literal Equations are Solve each of the following for s 9. C S V C N L 0
Word Problems General Strategy of Solving Word Problems 1.. 3. 4. Example: 1. Twice the difference of a number and 8 is equal to three times the sum of the number and 3. Find the number. Step 1: Step : Step 3: Step 4:. To make an international call, you need the code for the country you are calling. The codes for Belgium, France and Spain are three consecutive integers whose sum is 99. Find the code for each country. Step 1: Step : Step 3: Step 4: 1
Practice: 15. The sum of twice a number and 7 is equal to the sum of a number and 6. 16. If ¾ is added to three times a number, the result is ½ subtracted from twice the number. 17. The room numbers of two adjacent classrooms are two consecutive even numbers. If their sum is 654, find the classroom numbers 18. A 40-inch board is to be cut into three piece so that the second piece is twice as long as the first piece and the third piece is 5 times as long as the first piece. Find the lengths of all three pieces.
Formulas A lw P l w P a b c 1 A bh V lwh A r P r d rt I PRT 9 F C 3 5 3
Further Problem Solving Solve problems involving Percents Increase Percent Increase = Decrease Percent Decrease = 1. Nordstrom s advertised a 5% off sale. If a London Fog coat originally sold for $56, find the decrease in price and the sale price 3. How many cubic centimeters (cc) of a 5% antibiotic solution should be added to 10cc of a 60% antibiotic solution to get a 30% antibiotic solution? 4
5. A jet plane traveling at 500mph overtakes a propeller plane traveling at 00mph that had a - hour head start. How far from the starting point are the planes? 6. Karen invested some money at 9% annual simple interest and $50 more than that amount, at 10% annual simple interest. If her total yearly interest was $101, how much was invested in each? 5
Class 3: Quadratics Objectives: Perform Operations with Complex Numbers Solve Quadratic Equations by any method Solve Polynomial Equations by factoring Solve Radical Equations. Solve Equations with Rational Exponents Solve Equations involving Absolute Values Solve linear and Absolute Value Inequalities The Imaginary Unit is A complex Number is Complex Conjugate Operations with Complex Numbers o Powers of Imaginary Numbers. Practice 1. Perform the indicated Operation i 5 i 7 6
Addition/Subtraction: Subtract: 7 5i 9 11i Multiplication. Multiply: 9 45i5 i o Division. Practice 5: 8 5 i 8 5i 7
Quadratic Equation A Quadratic Equation Zero Factor Theorem: Example: x 5x14 0 Factoring : Solving Quadratic Equations by Factoring: 1.. 3. 4. 5. Solve: 5x 0x 60 0 Square Root Property If x afora 0 x a Example: x 49 0 x = Solve by the square root property: ( x 4) 36 x = 8
Completing the Square To complete the square Example x 8x1 0 General Strategy for Completing the Square 1.. 3. 4. 5. Example: x x 4 9
Quadratic Formula The Quadratic Formula Solve by using the Quadratic Formula: x x 5 0 Practice 6: x 8x 3 The Discriminant Discriminant Number of Solutions Practice: Use the discriminant to find the # of solutions 7. x x 3 0. x x 0. x x 1 0 30
Other Equations Radical Equation Give at least two Examples Domain Find the domain x 4 9 Strategy on Solving Radical Equations containing nth Roots 1.. 3. 4. Solve the equation: 15 3x17 x 31
Equations with Rational Exponents: Equations with rational exponents are Rewrite as radical: m n a Strategy on Solving Equations with Rational Exponents 1.. 3. 4. Solve the following equations for their real solutions. 10. 5 3 8x 4 0 11. 1 3 3 x 4x 6 3
Equations Involving Absolute Values x Solve: 5 3x 7 6 1 Inequalities Linear Inequalities Solve the following inequality 5x 11 6 33
Absolute Value Inequalities If u is an algebraic expression and c is a positive number, then u c And u c Solve the following inequality 7 x 8 14 35 34
Class 4: Function and Their Graphs Objectives: Identify and Graph Functions Identify Domain and Range Identify Characteristics of Functions Calculate the slope of a Line. Write and find the point-slope and Slope intercept of the equation of a line Solve Equations involving Absolute Values A Relation is Domain Range Functions is Domain Range There are four possible ways to represent a function: List them below: 1.. 3. 4. 35
Example: - Verbally: The area of a square plot of land is equal to the square of the length of the lot. - Numerically: (0,0), (1,1), (,4), (3,9), (4,16) Or Length 0 1 3 4 Area 0 1 4 9 16 - Visually: - Algebraically: A(s) = s Notation: A function f of x is represented as: x represents y - represents The Graph of a Function Determining whether a relation is a function o Numerically Practice: Determine if the following examples are functions. If not, explain. 1. In the following ordered pairs the first element represents Number of hours worked and the second element represents Total pay. (0, $0) (1, $7.50) (, $15.00) (3, $.50) (4, $30.00) (5, $37.50) (6, $45.00) (7, $5.50) (8, $60.00). The first element of each ordered pair is Student First Name and the second element of each ordered pair is Number of Math Courses Taken. 1. (Peter, ). (Jackie, 0) 3. (Marian, ) 4. (Tammy, 3) 5. (Jess, 1) 6. (Jackie, 1) 7. (John, 3) 8. (Joe, ) 9. (Ron, 0) 36
o Algebraically To determine if an equation is a function Practice: Determine if the following equations define y as a function of x. 3 3. x y 14 5. xy 3y 4 3 4. x y 14 o Visually The Vertical Line Test : Practice: Determine if y is a function of x. 6. 7. 8. 37
Graph of a Function Arrows indicate A closed dot indicates An open dot, indicates Finding Domain and Range Practice: Find the Domain and Range in each of the following cases: o Numerically 7. (0, 1650), (10, 1750), (0, 1860), (30, 070), (40, 300), (50, 560), (60, 3040), (70, 3710), (80, 4450), (90, 580) Domain Range o Visually 8. Domain: Range: http://lima.osu.edu/people/iboyadzhiev/geogebra/domain_range.html 38
Algebraically The Domain of any polynomial function is Exceptions: - - Give an example of each of the above exceptions: Evaluating Functions Same process as evaluating an algebraic expression 9. Example: Consider the function a. f ( 3) f ( x) x 5x 3. Evaluate the following: b. f( h ) c. f( h ) 39
Piece-Wise Functions 10. Practice: Graph the following function. x 3, if x 0 f ( x) 4, if x 0 x 6, if x 0 Domain: Range: Evaluate f (-)= ------------------------------------- f(3)= ---------------------------------------------- Characteristics of Functions: DOMAIN - RANGE - MAX/MIN - Increase/Decrease - X-INTERCEPTS - Y-INTERCEPTS - Odd Even- 40
Linear Functions and Slope Write the General Form of the Equation of a Line: Write the equation of a horizontal line: Write the equation of a vertical line: Slope is Write the formula used to find the slope of a line Write the slope -intercept form of a linear equation and state what each part represent. Write the Point-Slope form of the Equation of Line: State the appropriate slope for each of the following cases: Vertical Line Horizontal Line 11. Fill in the appropriate slope for each of the lines below: Tilts Upward Tilts Downward Horizontal Vertical y y y y x m m m m 41
Practice: For the each of the following find the slope of the line through the points: 1. (-,-5),(0, -), (4,4), (10, 13) 13. (-,1), (3,5) 14. State the slope of each of the lines given by the equations below: x a. y = 3x - 5 b. y 4 7 m = m = 15. Find the equation of the line that goes through the points (-,3) and (-5, -1). 4
Class 5: Average Rate of Change and Transformations Objectives: Calculate Average Rate of Change Calculate the Difference Quotient Recognize Graphs of Common Functions Use transformations to graph Functions Average Rate of Change: Practice: For each of the following functions, find the average rate of change. 1. f x x x x to x ( ) 1 3 6 Difference Quotient Difference Quotient Example: Calculate the difference quotient for the function f x x x ( ) 7 11 43
Basic Functions Linear f(x)= Quadratic f(x)= Domain: Range: Cubic f(x)= Domain: Range: Rational f(x)= Domain: Range: Radical f(x)= Domain: Range: Exponential f(x)= Domain: Range: Domain: Range: 44
Transformations of Functions Transform ation Equation Description Vertical translation y f ( x) c y f ( x) c f ( x) x g( x) x 3 Horizontal y f ( x c) f ( x) x g( x) x 4 translation y f ( x c) y f ( x) f ( x) 3 x Reflections y f ( x) h( x) 3 x Vertical Stretching/ Shrinking y cf ( x) f ( x) x g( x) x Horizontal Stretching/Sh rinking y f (c x) f ( x) x 1 g( x) x 45
Practice: Describe the change in the graph of the function and then graph it. f ( x) x for each of the following transformation, a. f x ( ) x b. f ( x) x c. f ( x) x d. f ( x) x e. 1 f ( x) x 46
Class 6: Composition and Inverse Functions Objectives: Combine functions using the algebra of functions Determine domain of Functions and of composite functions Write Functions as Compositions Verify inverse functions Find the Inverse of a Function Determine if a function has an inverse Graph a Function and its Inverses Domain of Functions Domain: Practice: For each of the following functions, find the domain.. 4 f ( x) 6x x x 5 x 3. hx ( ) 3x 19x6 4. k( t) x 16 47
Algebraic Operations with Functions Four algebraic operations that we do with polynomial functions are: Practice: Perform the indicated operations for the following functions. f x x g x x x h x x x 3 ( ) 3, ( ) 3, ( ) 5 6, 5. f(x)+h(x)= 6. h(x)-g(x)= 7. f(x) g(x)= 8. gx ( ) f( x) 48
Composition of Functions The Composition of the function Practice: Perform the indicated operations for the following functions. f x x g x x x ( ) 4 ( ) 3 4 9. Compose f ( x) g( x) 10. Compose g f x (1) 49
Inverses The Inverse of a function f x Example: Determine if f ( x) x 6 and gx ( ) 3 are inverse functions 50
Finding Inverse Functions Steps To find Inverse Functions 1.. 3. 4. 5. Example: Find the inverse of f ( x) 3x 1 51
Existence of Inverse Functions Does every function have an inverse? How do we determine if a function has an inverse? Algebraically: Example: Graphically: The horizontal Line Test: Practice: Determine if the following functions have an inverse 5
Class 7: Distance and Midpoint Formulas; Circles Objectives: Find the Distance between two points. Find the midpoint of a line segment Write the standard form of a circle s equation Give the center and radius of a circle whose equation is in standard form Convert the general form of a circle s equation to standard form The Distance Formula The midpoint Formula 1. Plot the points A(4, 6), B(-3, ), and C (1,-5) on a coordinate system and connect them in order to find a triangle. a) Calculate the lengths of the three sides of the triangle. 53
Circles A Circle is Radius is The Standard Equation of a circle is The General Form of the Equation of a Circle is Practice: Write the standard equation for the circle in each of the following cases;. Center (-3, 5), r = 3 Practice: Give the center and radius of the circle described by the following equations: 3. x y 5 4 6 Practice: Complete the square and write the equation in standard form, then graph it and use it to identify the domain and range. 4. x y x y 8 4 16 0 54
Class 8 & 9 Review and Test 1 Summary/Questions 55
Class 10: Angles and Their Measurements Objectives: Define and draw angles Convert angles from Degrees to Radians Convert angles form Radians to Degrees Use Right Triangles to Evaluate Trigonometric Functions Definitions: Draw each of the following: Line: Line Segment: Ray : Angle: Standard Position: Positive Angles Negative Angles Quadrantal Angles 56
Measuring Angles We measure angles by By Degrees ( ) One Revolution We can classify angles by degrees: Acute angle Right angle Obtuse Angle Straight angle Practice: Classify the following angles: 1. 15 -. 160-3. 65-4. 90-5. 45-6. 180-57
By Radians Central Angle: One Radian Radian Measure Example: Find the measure of the angle θ that intercepts an arc of length 15 inches in a circle of radius 6 in. Relationship between Degrees & Radians Conversions: To convert degrees to radians, To convert from radians to degrees, Practice: Convert from radians to degree or degrees to radians as necessary 7. 30 = 8. 5 3 radians 58
Fill the circle with the degree and radian measure Practice: State the quadrant each angle is and then draw the angle in standard position. 9. 3 5 10. 3 4 59
Trigonometric Functions Trigonometric Functions Sin θ = Csc θ = Cos θ = Sec θ = Tan θ = Cot θ = Do the values of the trigonometric functions depend on the length of the sides of a triangle? 11: Find the value of each of the six trigonometric functions of θ for the following triangle. a=5 c b=1 θ 60
Special Angles 30 45 60 or,, 6 4 3 45 or 4 sin 4 csc 4 cos 4 sec 4 tan 4 cot 4 30 or and 0 or 6 3 Example: sin 3 sin 6 cos 3 cos 6 tan 3 tan 6 61
Special Identities Reciprocal Identities: sin cos tan csc sec cot Pythagorean Identities: Example: Given that 1 sin and θ is acute, find cos Practice: Use identities to find the trigonometric function. 1. Find sin if 7 cos 13. Find tan if 8 6 sin 7 6
Co-terminal Angles Co-terminal angles Example: Practice: Find a positive angle less than π that is co-terminal with each of the following. 14. 400 15. -135 63
Class 11: Trigonometric Functions of any Angle Objectives: Trigonometric functions of any angle/definition Use the signs of the trigonometric functions Reference Angle Applications of Trigonometric Functions Definition of Trigonometric Functions of any Angle: Sin θ = Csc θ = Cos θ = Sec θ = Tan θ = Cot θ = Example: Let P ( 3, 4) be a point in the terminal side of θ. Find the value of the six trig. functions. 64
The Signs of the Trigonometric Functions The table summarizes the signs of the Quadrant II Quadrant I trigonometric functions Quadrant III Quadrant IV Here is an easy way to remember: I II III IV Example: Given 1 tan and cos 0, find sinθ and secθ. 3 65
Quadrantal Angles: Lets find the values of trigonometric functions for the quadrantal angles. y θ 0 π sinθ 90 π/ 180 π 70 3π/ cosθ tanθ Values of Special Angles θ sin θ 30 45 0 0 6 4 3 90 180 70 3 cos θ tan θ 66
Reference Angles Reference Angles Example: Finding Reference Angles: If Example: Find the reference angle of θ = 10 7 4 67
Why do we need to know Reference Angles? Example: Find the exact value of 4 cos 3 Practice: Use identities to find the trigonometric function. 1. Find the exact value of tan( 10 ). Find the exact value of 11 csc 4 68
Class 1: Trig Functions of Real Numbers & their Graphs Objectives: Trigonometric functions of real numbers Recognize Domain and Range of Sin and Cos functions Use of Even and Odd trigonometric Functions Use of Periodic Properties Graph the sine and cosine functions and their transformations Trigonometric Functions of Real Numbers Cycles govern many aspects of our lives such as sleep patterns, seasons, tides etc. All follow regular, predictable cycles. In this section we are going to see why trigonometric functions are used to modes such phenomena. Until now we have considered trigonometric functions of angles. To define trigonometric functions of real numbers rather than angles we use a unit circle. Unit Circle: Definition of Trigonometric: Sin θ = Cos θ = Tan θ = Csc θ = Sec θ = Cot θ = Example: P 3 4, 5 5 69
The Graph of Sine y sin x To graph a function x 0 6 3 5 3 6 7 4 3 5 11 6 3 3 6 Y = sinx Sketch a neat plot of the graph you got below: Characteristics of the basic function y = sinx Domain: Range: Period: Odd/Even: x-intercepts: Max/Min: 70
General Equation of Sine Function: Amplitude Period Phase Shift Vertical Shift Example: Determine the period, phase shift, and amplitude for y3sin x and graph it. 3 71
The Graph of Cosine y cos x We are going to graph y = cosx also by listing some points on the graph. To graph a function x 0 6 3 5 3 6 7 4 3 5 11 6 3 3 6 Y = cosx Sketch a neat plot of the graph you got below: Characteristics of the basic function y = cosx Domain: Range: Period: Odd/Even: x-intercepts: Max/Min: General Equation of Cosine Function: 7
Example: Determine the period, phase shift, and amplitude for y 4cos x and graph the function. Practice: Graph the function y x cos 4 3 73
The Graph of Tangent y tan x We are going to graph y = tanx also by listing some points on the graph. To graph a function x 0 6 3 5 3 6 7 4 3 5 11 6 3 3 6 Y = tanx Sketch a neat plot of the graph you got below: Characteristics of the basic function y = tan x Domain: Range: Period: Odd/Even: x-intercepts: Max/Min: General Equation of Tangent Function: 74
Class 13: Inverse Trigonometric Functions & Applications Objectives: Understand and use the inverse Sine, Cosine and Functions Use calculators to evaluate inverse trigonometric functions Find exact values of composite functions with inverse trigonometric functions Solve a Right Triangle Application Of Trigonometric Functions RECALL: The graph of the trigonometric functions are below: Sine Cosine Tangent If we restrict the domain of these functions we will get the following graphs: Domain: Range 75
The graph of the inverse trigonometric functions are below: Arcsine Arcosine Arctangent Function: Domain: Range Example: Find the exact value of each of the following: sin 1 = 3 arccos = 1 sin 3 = 1 cos 1 = 1 tan (0) = arctan 3 = 3 76
Properties of Inverse Functions 1 sin(sin x) 1 sin (sin x) 1 cos(cos x) 1 cos (cos x) 1 tan(tan x) 1 tan (tan x) Using Inverse Properties: Evaluate sin sin 4 1 1 tan(tan ( 5)) 1 cos(cos ) 77
Solving Right Triangles Solving a Triangle means A C B 1. Let A = 6.7 and a = 8.4. Solve the right triangle shown below rounding to two decimal place Find x to the nearest whole unit. 78
Angle of Elevation and Angle of Depression Angle of Elevation is Angle of Depression is Example : From a point on a level ground 80 ft from the base of Eiffel Tower, the angle of elevation is 85.4. Approximate the height of the Eiffel Tower to the nearest foot. 79
Class 14 Trigonometric Identities Objectives: Use various methods to verify Trigonometric Identities 1.. 3. 4. Example: Verify csc x tan x sec x 80
Class 15 Trigonometric Equations Trigonometric Equation Steps in Solving Trigonometric Equations 1.. 3. 4. Practice: Solve the following equations: 1. 5sin x3sin x 3 81
. sin 3sin 1 0 0 x x x 3. sin x tan x sin x 0 x 4. sin 3cos 0 0 x x x 8
Class 16: The Law of Sines & The Law of Cosines Objectives: Use the Law of Sines and Cosines to solve oblique triangles Solve applied problems using the Law of Sines and Cosines Oblique Triangle Note: The Law of Sines: Example: Solve the triangle ABC if A = 40, C =.5 and b=1. 83
Practice: 1. Solve the triangle ABC if A = 57, a = 33 and b = 6.. Solve the triangle ABC if A = 35, a = 1 and b = 16. 84
The Law of Cosines When given all three sides Example: Solve the triangle with A=10, b=7 and c=8. Example: Solve the triangle ABC if a = 8, b = 10 and c = 5. 85
Class 17 & 18 Review and Test Summary/Questions 86
Class 19: Quadratic Functions Objectives: Recognize Characteristics of Parabolas Graph Parabolas Determine a Quadratic Function s Max/Min Value Solve problems involving a quadratic function s max/min value. Basic quadratic function Vertex Standard Form of a Quadratic Equation o o o Practice: Identify the vertex and axis of symmetry of each parabola below. 1. f x x ( ) 3 5 To graph a quadratic Function in Standard form Practice: Graph the following quadratic functions a. f x x ( ) 3 1 87
General Form of a Quadratic Equation Vertex Practice: Find the vertex for f x x x ( ) 3 10 Practice: Graph the following quadratic functions f x x x ( ) 4 3 88
Class 0: Polynomial Functions & Division of Polynomials Objectives: Identify polynomial functions Recognize characteristics of graphs of Polynomial Functions Determine end behavior Identify zeroes and their multiplicities Use synthetic division to divide polynomials Use the Rational Zero Theorem to find possible rational zeros Find zeros of Polynomial Functions Polynomial Functions Practice: Which of the following functions are polynomial functions? 7 3 a. f ( x) 3x 5x x b. c. 4 f ( x) x 5x 8x 3 3 ( ) 7 f x x x x Graphs of Polynomial Functions Smooth Continuous End Behavior If the degree is odd If the degree is even If the leading coefficient is positive If the leading coefficient is negative 89
Example: Determine the end behavior of f x x x x 3 ( ) 5 7 5 Practice: Determine the end behavior of f x x x x 6 5 ( ) 7 1 Zeroes of Polynomial Functions - - - - - Example: Find the zeros of the polynomial functions below f ( x) x 3 x 1 f x x x x 3 ( ) 7 4 8 90
Synthetic Division. x 5x 5 x 5 3. 5 3 6x x 4x 5x 5 x Use synthetic division to evaluate f (1) for f x x x x 3 ( ) 4 7 91
Zeros of Polynomial Functions Rational Roots of Polynomial Functions Example: List all the possible rational zeroes of f x x x x x 4 3 ( ) 3 11 19 6 Practice: Find all possible rational zeros and use long/synthetic division to test then and find the 3 actual ones for f ( x) x x 4x 4 9
Class 1: Rational Functions Objectives: Find the Domain of Rational Functions Identify Vertical Asymptotes Identify Horizontal Asymptotes Applications of Rational Functions Rational Function Give at least two Examples Domain To find the domain Example: Find the domain of f( x) x x9 ( x3)( x 16) The Basic Rational Function is Domain : Range : Y int : X int : Asymptotes : DirectionalLimits : 93
Asymptotes An Asymptote is Finding Asymptotes: - Vertical - Horizontal - Slant Practice: Find all Asymptotes for each of the following functions x 1. f( x) x x6. f( x) 3 x 7 x 3 3. fx 3 x 1 3 3x x 94
Characteristics and Graphs of Rational Functions What do we need to know about R.F.? - - - - - Example Give the characteristics and sketch a graph for each of x 1 4. f( x) x 1 Domain: Range: x-int: y-int: HA: VA: SA: D. Limits: 95
Variation Direct Variation: Inverse Variation Joint Variation 1. The amount of gas that a helicopter uses is directly proportional to the number of hours spent flying. The helicopter flies for 3 hours and uses 4 gallons of fuel. Find the number of gallons of fuel that the helicopter uses to fly for 6 hours.. The weight of a body above the Earth's surface varies inversely with the square of the distance from the center of the Earth. If a certain body weighs 55 pounds when it is 3960 miles from the center of the Earth, then how much will it weigh when it is 3965 miles from the center of the Earth? 96
Class : Exponential and Logarithmic Functions Review: Basic Laws of Exponents: Write the exponential rules for each of the following (See pg. 4) x y x 1 a a ab a x a 0 a x a y a b y x a a x a x y Simplify the following: 7 5 3 4 3 ( ) (16) 3 ( xy) 3 4 5 1 3 3 5 3 (3) (7) 1 3 97
Exponential Functions The exponential function o y-intercept: o Domain: o Range: o Asymptotes? o Inverse In college, we study large volumes of information that, unfortunately we do not often retain for 0.5 x very long. The function f ( x) 80e 0 describes the percentage of information that a person can be expected to remember x weeks after learning it. a. Let x = 0 and give the value of f(0) b. Let x = 5 and determine the value of f(5) accurate to the nearest ten thousandth 98
Compounding Simple Interest Compound Interest The Natural Base e : Continuous Compounding: Laura borrows $500 at a rate of 10.5%. Find how much Laura owes at the end of 4 years if: a. The interest is compounded yearly b. The interest is compounded quarterly c. The interest is compounded monthly d. The interest is compounded continuously e. Which option would yield the most interest, 10.5% compounded monthly for 4 years or 9% compounded continuously? 99
Logarithmic Functions The Logarithmic Function A logarithm as an exponent: Write the Basic Laws of Logarithms below: Write the following in its equivalent exponential form: 1. log 16. log 64 4 6 3. log 6 16 y 4. log 5 15 y Write the following in its equivalent logarithmic form: 3 5. 8 6. 13 x 7. 5 3 1 15 y 8. 7 00 100
Natural Logarithm The Natural Logarithmic Function Properties of ln(x). Write the properties of the natural logarithm below: Simplify the following 1. 6 ln e. 1 3. ln e 6 4. 7 log 10 log53 10 5. ln15 e 6. e ln 7x 101
Logarithmic Functions o y-intercept: o Domain: o Range: o Asymptotes? o Inverse Example: Find the domain of the following functions: a. f ( x) ln( x ) b. f ( x) log 3x 6 The percentage of adult height attained by a girl who is x years old can be model f ( x) 6 35log( x 4) where x represents the girl s age and f(x) represents the percentage of her adult height. a. Approximately what percentage of her adult height has a girl attained at age 13? b. Approximately what percentage of her adult height has a girl attained at age 16? 10
Class 3: Exponential and Logarithmic Equations and Logistic Growth Exponential Equation Give at least two example: Examples: Solve the following equations 1. 4 x 1 64. x 3 1 1 7 3. x e x e 4 1 103
Steps in Solving Exponential Equations 1.. 3. 4. Example: Solve for x 5x 3e 1977 4. 0.00 x 5e 9 1 104
Logistic Growth Logistic Model 100, 000 7. The logistic growth function f() t 1 500e describes the number of people, f(t), who t have become ill with influenza t weeks after its initial outbreak in a particular community. a. How many people became ill with the flu when the epidemic began? b. How many people were ill by the end of the fourth week? c. What is the limiting size of the population that becomes ill? 105
Properties of Logarithms - Product Rule: log ( x )( x 3) log( x ) log( x 3) Example: - Quotient Rule: Example: (x 3) log log(x 3) log( x5) ( x 5) - Power Rule: Example: 7log( x3) log( x 3) 7 106
Class 4 & 5 Review and Test 3 Summary/Questions 107
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